Formal methods for the description of spatial relations can be based on mathematical theories of order. Subdivisions of land are represented as partially ordered sets (posets), a model that is general enough to answer spatial queries about inclusion and containment of spatial areas. After a brief introduction to the basic concepts of posets and lattices, their applications to modeling spatial relations and operations for spatial regions in terms of containment and overlay are presented. An interpretation is given for new geographic elements that are created by the completion from a poset to a lattice. It is shown that a novel approach to characterize certain topological relations based on a lattice of a simplicial complex is a model for spatial regions that combines both topological and order relations and allows spatial queries to be answered in a unified way.

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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper

In this research report, we present a new process for defining and building the set of configurations of Marching-Cubes algorithms. Our aim is to extract a topologically correct isosurface from a volumetric image. Our approach exploits the underlying discrete topology of voxels and especially their connectedness. Our main

This is a subjective essay, and its title is misleading; a more honest title might be how i write mathematics. It started with a committee of the American Mathematical Society, on which I served for a brief time, but it quickly became a private project that ran away with me. In an effort to bring it under control I asked a few friends to read it and criticize it. The criticisms were excellent; they were sharp, honest, and constructive; and they were contradictory. "Not enough concrète examples " said one; "don't agrée that more concrète examples are needed " said another. "Too long" said one; "maybe more is needed " said another. "There are traditional (and effective) methods of minimizing the tediousness of long proofs, such as breaking them up in a séries of lemmas " said one. "One of the things that irritâtes me greatly is the custom (especially of beginners) to présent a proof as a long séries of elaborately stated, utterly boring lemmas" said another. There was one thing that most of my advisors agreed on; the writing

Abstract—Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimum-cost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan’s solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data. Index Terms—3D image segmentation, minimal surfaces, shortest paths, Dijkstra’s algorithm, boundary operator, total unimodularity, linear programming, minimum-cost circulation network flow. Ç 1

Abstract. For a given pair of maps f, g: X → M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg: H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X such that f(x) = g(x). 1. Introduction. Consider the Fixed Point Problem: “If X is a topological space and g: X → X is a map, what can be said about the set Fix(g) of points x ∈ X such that g(x) = x? ” The Coincidence Problem is concerned with the same question about two maps f, g: X → Y and the set Coin(f, g) of x ∈ X such that f(x) = g(x).

Abstract. A Lefschetz-type coincidence theorem for two maps f, g: X → Y from an arbitrary topological space to a manifold is given: Ifg = λfg, the coincidence index is equal to the Lefschetz number. It follows that if λfg ̸ = 0 then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X, Y manifolds and (ii) f with acyclic fibres.

A polynomial algorithm was given by the authors and Bernardinello for synthesizing pure weighted Petri nets from finite labeled transition systems. The limitation to pure nets, serious in practice e.g. for modelling waiting loops in communication protocols, may be removed by a minor adaptation of the algorithm, working for general Petri nets fired sequentially. The rule of sequential firing reduces also the expressivity of Petri nets, since it forces a concurrent interpretation on every diamond. This limitation may also be removed by leaving sequential transition systems and lifting the algorithm to step transition systems, which amounts to extract the effective contents of the coreflection between Petri step transition sytems and general Petri nets established by Mukund. By the way, the categorical correspondences between transition systems or step transition systems and nets are re-examined and simplified to Galois connections in the usual setting of ordered sets.